Probability distribution Dr. Deshi Ye College of Computer Science, Zhejiang University

Outline  Chebyshev’s Theorem  The Poisson distribution & process  The Geometric Distribution  The Multinomial distribution  Simulation

4.5 Chebyshev’s theorem  EX. The number of customers who visit a car dealer’s showroom on a Saturday morning is a random variable with mean 18 and standard variance 2.5.  Question: What probability can we assert that there will be more than 8 but fewer than 28 customers?

Chebyshev’s theorem  Theorem 4.1. If a probability distribution has mean and standard deviation, the probability of getting a value which deviates from by at least is at most Symbolically

Proof. Since

Corollary

EX  The number of customers who visit a car dealer’s showroom on a Saturday morning is a random variable with mean 18 and standard variance 2.5.  Question: What probability can we assert that there will be more than 8 but fewer than 28 customers?

Solution of EX  Let X be the number of customers. Hence k = 4

4.6 Poisson Distribution  Poisson distribution: often serves as a model for counts which do not have a natural upper bound.  Poisson distribution

Comparing Poisson and binomial  It is known that 5% of the books bound at a certain bindery have defective bindings. Find the probability that 2 of 100 books bound by this bindery will have defective bindings using  A) the formula for the binomial distribution  B) the Poisson approximation to the binomial distribution

Solution  A) x=2, n=100, p=0.05 hence  B) x=2, for the Poisson distribution, we

Poisson approximation to binomial distribution EX: P129

Poisson distribution  Mean and variance of Poisson distribution Proof.

4.7 Poisson Processes  On average 0.3 customer/min at a cafeteria,  Question: then what is the probability that 3 customers will arrive in 5- minute span?

Random process  Random process: is a physical process that is wholly or in part controlled by some sort of chance mechanism.  Characterize: time dependence, certain events do or do not take place at regular intervals of time or throughout continuous intervals of time.

 Goal: find the probability of x success during a time interval of length T.  Divide T into n equal parts of length ∆t T=n ∆t 1) The prob. of a success during a very small interval ∆t is given by a∆t 2) The prob. of more than one success during a small time interval ∆t is negligible. 3) The prob. of a success during such a time interval does not depend on what happen prior to that time.

By Poisson probability  When n is large enough the probability of x success during the time interval T is given the corresponding Poisson distribution with the parameter

Solution of EX  Solution: Consequently

4.8 The Geometric Distribution  Suppose that a sequence of trials we are interested in the number of the trials on which the first success occurs.  Geometric distribution

Mean and variance  Mean of geometric distribution  Variance of geometric distribution

4.9 The multinomial distribution Expansion of

4.10 Simulation  Simulation can be useful tools for finding approximate probabilities for situation in which the actual probabilities are too difficult to calculate.  Approximation will typically be better the more repetitions you perform.

Activities  Divide students into groups such that each group has 3 students and also have 3 cards. 1) 3 cards written their own names. 2) Randomly shuffle 3) Each one pick cards, count number of matches TrialNameWhose nameNumber of matches 1Amy Barb Carol Barb Amy Carol 1

Theoretical analysis  XYZ, ABC, match X->A, Y->B, C->Z ABC ACB BAC BCA CAB CBA Probability /3 ½ 0 1/6

4-Person Analysis prob9/248/246/2401/24

Case study  The Coupon Collector’s problem